demand perspective of species invasions and coexistence: applications to biological control

Ecological Modelling 106 (1998) 27 – 45

A supply/demand perspective of species invasions and coexistence: applications to biological control Sebastian J. Schreiber *, Andrew P. Gutierrez Di6ision of Ecosystem Science, College of Natural Resources, Uni6ersity of California, Berkeley, CA 94720, USA Accepted 12 September 1997

Abstract Using a physiologically-based model with a supply/demand driven functional response, the dynamics of a species invading a tritrophic food chain are analyzed leading to four conclusions. First, it is shown that a trophic stack may establish via a sequence of invasions, approach a stable steady-state and collapse following a perturbation that significantly alters the ratio of biomass between adjacent trophic levels. Second, when two consumers compete for a common prey, the outcome is determined by a simple rule: the competitor with the lowest metabolic compensation point (the ratio of prey biomass to total consumer demand at which prey assimilation exactly compensates for metabolic costs) displaces the other. Third, intraguild predation can mediate coexistence only if the intraguild prey’s metabolic compensation point is lower than the intraguild predator’s metabolic compensation point. Fourth, when two consumers share a prey and a predator, a simple rule determines dominance: the consumer with the lowest ecological compensation point excludes the other. The ecological compensation point of a consumer is defined to be the ratio of prey biomass to consumer demand when the prey – consumer – predator food chain is at equilibrium. To illustrate the utility of the analysis, several intensively studied biological control programs are examined by parameterizing the model using life table statistics, field observations and mass scaling laws. Using our invasion criteria, we construct assembly diagrams that describe how invasions result in a transition from one persistent subsystem to another. In all cases, there exist paths within these theoretical diagrams that correspond to the observed case histories. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Food webs; Persistence; Ratio-dependence; Population dynamics; Metabolic pool paradigm; Biological control; Biological invasions

1. Introduction * Corresponding author. Present address: Department of Mathematics, Western Washington University, Bellingham, WA 98225. E-mail: [email protected]

To successfully invade, a species needs to be physiologically adapted for the habitat of the

0304-3800/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 0 4 - 3 8 0 0 ( 9 7 ) 0 0 1 7 8 - 6

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S.J. Schreiber, A.P. Gutierrez / Ecological Modelling 106 (1998) 27–45

resident community, because if an abiotic factor of the habitat exceeds the invader’s limit of tolerance (Shelford, 1931) or the habitat lacks an essential resource, no invasion can occur. Given abiotic favorability, patterns of predation and competition within the community and interactions between the resident species the new arrival may infiuence the communities’ vulnerability to invasion (Elton, 1958; Drake, 1990). If the new arrival overcomes these obstacles and successfully invades, its impact on the resident community may produce dramatic shifts in species composition or go unnoticed as the species’ population increases slowly with a marginal impact on the resident community (Huffaker and Messenger, 1976). In this paper, we address these issues by extending a physiologically-based model of trophic interactions (Gutierrez et al., 1994) to multispecies interactions, analyzing the population dynamics of the model, and comparing its qualitative predictions with four intensively studied biological control programs. Mathematical models have been used to explore fundamental issues of coexistence and species displacement in food webs. However, when faced with a particular ecosystem, it has been difficult to determine (without extensive field work) which ecological relationships are relevant, what mechanisms are present and whether parameter estimates are accurate (Murdoch, 1994). Simple models can compound these difficulties by introducing parameters that lack biological meaning except in an abstract sense. At the other extreme, there are attempts to construct more complex models with extensive biological details for particular ecosystems (e.g. McCauley et al., 1990; Gutierrez et al., 1993). These models typically resist mathematical analysis and can only be understood by statistical and graphical examination of computer simulations. While answering specific questions about population growth and providing the basis for simplifying complex systems into an analytical framework, these models can not provide insight into general principles. To address the need for realism without sacrificing simplicity, paradigms of intermediate complexity that incorporate the physiology of energy acquisition and allocation have been integrated

into models that are amenable to analysis (Gutierrez, 1992; Yodzis and Innes, 1992). These physiologically based models have several advantages. First, allometric relations between body size and physiology provide estimates for many of the parameters (Thompson, 1942; Peters, 1983) and help determine feasible parameter combinations. Second, relationships between physiology and various abiotic factors (temperature, humidity, light intensity, salinity) are well documented (Peters, 1983; Gutierrez, 1996). Third, coupling models with these relationships offers insights into the impact of body size, physiology and abiotic factors on population dynamics (Yodzis and Innes, 1992). With these advantages in mind, we develop a physiologically-based model with a supply/demand driven functional response and use it to derive invasion and coexistence criteria for a species invading a tritrophic system consisting of a producer, an herbivore and a predator. These criteria allow us to develop hypotheses concerning the observed changes in species composition of several biological control programs. Biological control provides a convenient testing ground for the theory for three reasons. First, more often than not these natural enemies are released sequentially and a variety of outcomes has been observed: successive competitive displacements, invasions facilitating the resurgence or arrival of another pest, and guilds of coexisting natural enemies (Huffaker and Messenger, 1976). Second, many of the introduced natural enemies are specialists and, consequently, the food webs of these systems when restricted to strongly interacting species are relatively simple in form. Third, the short generation times of the species make extensive life history and physiological studies possible (Price, 1975).

2. Methods

2.1. The metabolic pool paradigm The metabolic pool paradigm views an organism as an energy processor (Phillipson, 1966): it acquires energy from its prey and allocates it to

S.J. Schreiber, A.P. Gutierrez / Ecological Modelling 106 (1998) 27–45

maintenance and production, with the end result that a portion of the ingested energy is converted into somatic or reproductive growth (Gutierrez and Wang, 1977; Yodzis and Innes, 1992). By viewing the population of a species as a meta-organism (Getz, 1993), the metabolic pool paradigm describes the biomass dynamics of a predator (m2) as follows 1 dm2 = u2F(m1, m2) −R2, m2 dt where m1 is the biomass of the prey, m2 is the predator’s biomass, F is the predator’s functional response, R2 is the base respiration rate per unit biomass of the predator (i.e. maintenance costs), and u2 is the efficiency of converting the assimilated prey biomass into predator biomass (Gutierrez, 1992). To complete the description of the predator’s biodynamics, we need to select an appropriate functional response. Traditionally, most models employ a purely prey dependent functional response. These models assume that predators only interact indirectly through prey exploitation: prey extraction by one predator depletes the prey available to another. While this assumption is reasonable at low predator densities, at higher densities predators are likely to interact with one another in a variety of ways (e.g. avoidance behaviors, contest competition, prey conditioning, shading, territoriality) (Price, 1975). A simple functional response model that attempts to tackle this issue is the ratio-dependent functional response (Schoener, 1973, 1976; Getz, 1984; Arditi and Ginzburg, 1989; Berryman, 1992). Despite the controversy surrounding it (Oksanen et al., 1992; Abrams, 1994; Diehl et al., 1994), this functional response is of particular interest as it complements the assumptions of the prey dependent functional response. To define the appropriate parameterization of the ratio-dependent functional response, we assume that any predator is in one of two states: actively searching for its prey or not. At the population level, predators change from one state to another at a rate determined by prey and predator densities. The per unit biomass demand of a predator, D2, is assumed to be the prime

29

motivation for search and, thus, determines the per-capita rate at which predators reinitiate searching. The per-capita rate at which searching predators discover the prey is assumed to be proportionate to the prey available per predator, m1/m2. We define this proportionality constant as a12, the apparency of the prey (species 1) to the predator (species 2). a12 is the rate a predator encounters its ‘share’ of the prey when actively searching and, thus, measures the susceptibility of the prey to discovery by the predator (Rhoades and Cates, 1976). Under the additional assumption that changes between these individual states occurs at a faster time scale than growth and respiration (see Metz and Diekmann, 1986), we get the following functional response F(m1, m2)= D2g

  a12m1 D2m2

(1)

where g(x)=x/(1+ x) (see Appendix A). Eq. (1) implies that under non-limiting conditions, the predator’s per unit biomass demand rate equals its maximal ingestion rate (i.e. D2g(a12m1/D2m2) approaches D2 as m1 gets large). Eq. (1) also implies that the apparency of the prey equals the maximal rate at which the population of predators extract their prey (i.e. D2m2g(a12m1/D2m2) approaches a12m1 as m2 gets large). These two observations imply that prey apparency and predator demand are dual to one another in the following way. Since, D2m2g(a12m1/D2m2)5 D2m2, predation is limited by the demand of the predator population. Alternatively, since D2m2g(a12m1/ D2m2)5 a12m1, predation is limited by the apparent prey density. Placing this into the context of a predator–prey model, we get: dm 1 a01m0 − R1m1 − = u1D1m1 dt a01m0 + D1m1 a12m1 D2m2 a12Rm1m+ D2m2 dm2 a12m1 D2m2 − 2 2 = u2D2m2 dt a12m1 +

(2)

where m0 is a constant flux of resource available to the prey. Since the resource flux is assumed to be constant, this predator–prey model best

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S.J. Schreiber, A.P. Gutierrez / Ecological Modelling 106 (1998) 27–45

describes plant–herbivore interactions where the plant receives a relatively constant flux of inorganic and physical resources (inorganic nutrients, water and light) or more generally predator – prey interactions where the prey is donor-controlled (Begon et al., 1990). In general, lower trophic levels can be added to create a multitrophic model whose first trophic level satisfies one of these hypotheses.

make two assumptions. First, we assume that predators ‘share’ prey proportionate to their demand. Hence, if species i consumes species j, then species i’s per unit biomass ‘share’ of the prey (species j ) is proportionate to

2.2. An extension of ratio-dependent models to food webs

where Pred( j ) is the guild of species that consume j (i.e. Pred( j )= {k such that ajk \ 0}). Thus, the denominator of Eq. (3) equals the demand of species j’s predator guild. A more general approach to prey partitioning as suggested by Schoener (1976) is discussed in Appendix A. Second, as in the previous section, we assume that each individual is in one of k+ 1 states (either actively searching or has just consumed one of its k prey) and the transitions between these states occur at a faster time scale than changes in the population state (Metz and Diekmann, 1986). At the population level transition rates between these states are as follows. The per-capita rate at which individuals reinitiate searching equals their demand (Di ). The per-capita rate at which a predator (say an individual of species i ) encounter one of its prey (say species j ) is proportionate to their ‘share’ of the prey (Eq. (3)). This proportionality constant equals the apparency of the prey to the predator (aji ). Under these assumptions, Appendix A shows that if species i consumes species j, then its functional response with respect to j is given by

A plausible extension of ratio-dependent models is presented here by modifying the functional response model in Eq. (1) to multiple predator – prey interactions. The extension is consistent with the ‘criteria for realism’ developed by Arditi and Michalski (1995) and Berryman et al. (1995), but differs from their approach in several ways. To define the extension, consider an ecological community consisting of n interacting species. We associate with each species (say species i ) the following parameters: its per unit biomass demand (Di ), its respiration rate (Ri ), its conversion efficiency (ui ) and the apparency of species j to species i (aji ). A summary of these parameters and important concepts associated with them is presented in Table 1. To increase realism, additional parameters can be included to capture more details of the physiology and behavior (e.g. species specific demand rates and conversion efficiencies, preference, switching parameters), but for the sake of clarity and simplicity, we refrain from including this additional complexity. To describe the flow of biomass within this community we Table 1 A summary of parameters and important concepts

Di mj %

k  Pred( j )

Di

Symbol

Apparency of species i to species j Assimilation efficiency Demand (maximal ingestion) rate Respiration rate Maximal assimilation rate Maximal productivity Production (tissue growth) efficiency

aij ui Di Ri Ai = ui Di Pi = ui Di−Ri Pi /Ai

(3)

Dkmk

aji mj %

k  Pred( j )

Terms for species i

,

Dkmk

:

1+

%

l  Prey(i )

ali ml %

k  Pred(l)

Dkmk

;

−1

,

where Prey(i ) is species i’s prey guild, the collection of species consumed by i (Prey(i )= { j such that aji \ 0}). The first term of this functional response, Di

aji mj %

k  Pred( j )

,

Dkmk

S.J. Schreiber, A.P. Gutierrez / Ecological Modelling 106 (1998) 27–45

is the rate at which species i encounters species j when actively searching. The second term,

:

ali ml

%

1+

l  Prey(i )

%

k  Pred(l)

Dkmk

;

−1

,

is the proportion of individuals of species i that are actively searching. The remaining individuals are handling, ingesting, digesting or resting following an encounter with a member of their prey guild. By summing over the prey guild of species i, the net assimilation rate of species i equals

ui Di mi

:

:

aji mj

%

j  Prey(i )

%

k  Pred( j )

aji mj

%

1+

j  Prey(i )

%

k  Pred( j )

;

Dkmk

Dkmk

;

·

−1

Subtracting respiratory costs and consumption by its predator guild, the biodynamics of species i are given by dmi = ui Di mi dt

:

:

aji mj

%

j  Prey(i )

;

%

k  Pred( j )

j  Prey(i )

−

:

aji mj

%

1+

%

j  Pred(i )

:

%

k  Pred( j )

Dkmk

%

k  Pred(i )

1+

%

l  Prey( j )

−Ri mi

%

k  Pred(l)

;

Dkmk

alj ml Dkmk

·

−1

aij mi

Dj mj

Dkmk

;

;

31

3. Results As a stepping stone to food webs of greater complexity, we analyze Eq. (4) to determine under what conditions a newly arrived species (species 4) can invade a tritrophic system consisting of a plant (species 1), a herbivore (species 2) and a predator (species 3), and what impact this invasion has on the resident species. We consider five variations of this invasion attempt. First, the invading species is a top predator that consumes the resident predator and does not interact with any other species. Second, the invading species is a predator that competes with the resident predator(s) for herbivore biomass. Third, it is a predator that competes for herbivore biomass with and also is attacked by the resident predator. Fourth (fifth), it is a herbivore (or plant) that competes for plant biomass (a limiting resource) and shares a common predator (herbivore) with the resident herbivore (plant). A summary of these five variations is depicted in Fig. 1. Note that in Fig. 1e, 0 refers to m0, the constant flux of resource available to plants. To simplify the presentation of the invasion criteria, we follow the bioenergetic nomenclature as used by Petrusewicz and MacFadyen (1970) and Odum (1971). We define the maximal productivity of species i, Pi, to be the sum of its maximal somatic and reproductive growth rates. For our model, Pi is realized when resources are non limiting and it equals ui Di − Ri. We define the maximal assimilation rate of species i, Ai, to be the maximal rate it can acquire and assimilate resources. For our model, Ai = Pi + Ri = ui Di (see Table 1).

3.1. Permanence in food chains

·

−1

(4)

Now that we have a model of food web dynamics, we need to analyze them to answer the question posed in the introduction.

Our study of ratio-dependent dynamics begins with the first variation, a top predator invading a tritrophic food chain. A necessary and sufficient condition for the invasion is that maximal assimilation rate of species 4 exceeds its respiration costs, u4D4 \ R4. As mentioned in our introduction, we assume throughout our discussion this condition is always met. Once a food chain has established, we show in Appendix B that no

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S.J. Schreiber, A.P. Gutierrez / Ecological Modelling 106 (1998) 27–45

Fig. 1. The five types of invasion into a tritrophic system ((0) a constant resource flux, (1) a plant, (2) an herbivore and (3) a predator by newly arrived species (4).

perturbation of populations’ densities can result in a deterministic collapse of the food chain provided that the maximal rate of biomass production at each trophic level exceeds the maximal rate of extraction by the next trophic level, Pi = ui Di − Ri \ai,i + 1,

(5)

for all i. This criterion for permanence was derived originally in Gutierrez et al. (1994) and indicates precisely when the isocline of the ith trophic level switches from a hump shape to an ‘L’ shape (see Fig. 2a)1. Numerical simulations and some analysis (Schreiber, 1996) suggest that if Eq. (5) is met for all species, then the unique equilibrium of the model is globally stable. However, when this inequality is not meet for a given trophic level (say i ), the food chain can collapse following a perturbation. For example, in Appendix B, we show that if Pi Bai,i + 1 −Ri + 1 − ai + 2, then all trophic levels above trophic level i − 1 are driven to extinction whenever the initial ratio of biomass between trophic levels i + 1 and i is too large. What makes this phenomena particularly interesting is that it can coincide with a 1 In Gutierrez et al. (1994), Ri does not appear in the equations since respiration took the form Ri m 1i + b with b\0.

stable equilibrium (see Fig. 2b) producing bistabilities. What this implies is that a trophic stack may establish via a sequence of invasions, approach a stable steady-state, and collapse following a perturbation that significantly alters the ratio of biomass between adjacent trophic levels.

3.2. Competition for limiting resources While food chains permit a vertical increase of species richness on a single resource, increasing species richness horizontally on a single resource is more difficult. Consider our second invasion scenario, a predator (species 4) that attempts to invade a tritrophic food chain by competing for herbivore biomass (Fig. 1b). In Appendix C, we show that species 4 can invade the equilibrium determined by species 1 and 3 if and only if P4 P3 a24 \ a23. (6) R4 R3 While Eq. (6) determines whether species 4 can invade, species 3 can invade the equilibrium determined by species 1, 2 and 4 if and only if the opposite inequality holds. Generically, only one of these inequalities can hold, defining the principle of competitive exclusion in physiological terms: the predator with the largest value of

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33

Fig. 2. The dynamics of a two trophic system (Eq. (2)). In (a), u1 =u2 =0.45; R1 =R2 =0.2; D1 =D2 =1; a01 =1.0; m0, the resource level available to species 1 equals 25; and a12 = 0.5. Since u1D1 −R1 \a12, the system is permanent. In (b), the parameters are as before except a12 =1.1 and R1 = R2 = 0.3 and there is a cone of extinction and a stable equilibrium. The heavy lines denote the isocline and lighter lines indicate trajectories of the differential equation.

P a R

(7)

is competitively dominant. It is worth noting that an exploitative model of resource competition also produces the same invasion inequalities. We refer the interested reader to Appendix D. Another way of looking at competitive dominance for these systems is by introducing the metabolic compensation point for a species: a critical value of a species’ supply/demand ratio (e.g. if i=3 or 4 then the supply/demand ratio is given by m2/Dimi ) where assimilation exactly compensates for metabolic costs and no production of biomass occurs (Gutierrez et al., 1984). This compensation point coincides with the supply/demand ratio when the predator – herbivore – plant system is at equilibrium and, therefore, equals the inverse of Eq. (7). Hence, our model suggests that the dominant competitor has the lowest metabolic compensation point. This simple rule parallels the R* rule (e.g. Volterra, 1928; Hsu et al., 1977; Tilman, 1990). The R* rule applies to exploitative competition and asserts that the dominant predator suppresses the prey to the lowest equilibrium density (denoted R*). This prey density is consistent with the dominant predator’s own maintenance but is below the maintenance requirements of all other competitors.

3.3. Asymmetric intraguild predation Intraguild predation occurs when two species compete for a common resource (say species 2) and one competitor (say species 3) attacks the other (species 4) (Polis and Holt, 1992). If the interaction is asymmetric (i.e. a43 \ 0 and a34 = 0), then species 3 is known as the intraguild predator and species 4 the intraguild prey (see Fig. 1c). To define the invasion criteria for species 4 we follow the lead of Petrusewicz and MacFadyen (1970) and introduce two terms. First, we define the rate at which biomass is lost to respiration and predation as the total biomass loss rate. For example, when the intraguild predator is at equilibrium in our model, the total biomass loss rate of the intraguild prey is equal to R4 + (P3/A3)a43 where P3/A3 equals the production efficiency (Odum, 1968; McNeil and Lawton, 1970; Peters, 1983) of the intraguild predator. Second, we define the maximal rate at which a species stores organic matter into tissue not consumed by other predators as the species’ total biomass growth rate. For example, when the intraguild predator is at equilibrium in our model, the total biomass growth rate of the intraguild prey equals its maximal productivity minus predation by the intraguild predator, P4 − (P3/A3)a34. In Appendix C, we show that the intraguild prey (species 4) invades the resident food chain if and only if

S.J. Schreiber, A.P. Gutierrez / Ecological Modelling 106 (1998) 27–45

34

4%s total biomass growth rate a24 4%s total biomass loss rate \

3%s productivity a . 3%s respiration 23

Equivalently, P3 a43 A3 P3 a24 \ a23 (8) P3 R3 R4 + a43 A3 Eq. (8) is a modification of Eq. (6) in that it includes the costs of intraguild predation on the invading species, (P3/A3)a34. Another way of looking at Eq. (8) is in terms of both species’ metabolic compensation point. Notice that for the intraguild predator the value of the metabolic compensation point depends on its resource (species 2 or 4). Using arguments presented in the previous section, a necessary condition for the invasion of the intraguild prey is that its metabolic compensation point is lower than the metabolic compensation point of the intraguild predator (i.e. Eq. (6) holds). Therefore, permanent coexistence can only occur when intraguild predation is sufficiently weak. P4 −

3.4. Top-down effects on in6asion Elton (1958) observed that in certain situations, the invasion of an exotic species is facilitated by a resident predator, and that often the invasion results in a new food chain replacing an old food chain. We begin our study of this issue with an invasion of a tritrophic food chain by an herbivore (see Fig. 1d). In Appendix C, we show that species 4 can invade the equilibrium determined by species 1–3 if and only if P3 P3 P2 − a23 a A3 43 A3 a14 \ a12 (9) P3 P3 R4 + a43 R2 + a23 A3 A3 where (P3/A3)a43 and (P3/A3)a23 are predation costs to species 2 and 4, respectively, and P3/A3 is the predator’s production efficiency. On the other hand, species 2 can invade the equilibrium determine by species 1, 3 and 4 if and only if the P4 −

opposite inequality holds. These observations imply that the following exclusion principle prevails: the herbivore with the largest value of total biomass growth rate plant apparency total biomass loss rate

(10)

displaces all other herbivores in the system. An alternative means to describe dominance in this system is by defining the ecological compensation point of an herbivore. Consider one of the food chains in our system. We define the ecological compensation point of an herbivore as its supply/demand ratio (m1/D2m2 and m1/D4m4 for species 2 and 4, respectively) when the plant–herbivore–predator system is at equilibrium. This differs from the metabolic compensation point as it includes the costs of predation. For a herbivore in this system, the ecological compensation point is determined by the inverse of Eq. (10). Hence, the dominant herbivore minimizes its ecological compensation point. Depending on the relative apparencies of the herbivore to the predator, the herbivore with the lower metabolic compensation point may or may not dominate (see field examples below). Finally, suppose the invading species is a plant (see Fig. 1e). Appendix C shows that it can invade if and only if P4 − ea42 P − ea12 a \ 1 a , R4 + ea42 04 R1 + ea12 01

(11)

where a0i is the apparency of the plants’ resources to plant species i and P3 a A3 23 A2

P2 − e=

is the production efficiency of the herbivore corrected for predation costs. In the absence of predation, Eq. (11) reduces to Eq. (9). Such a system can have multiple switches in dominance. For example, suppose the metabolic compensation point of plant 1 is lower than the metabolic compensation point of plant 4 and its ecological compensation point with herbivory and no predation is higher than that of 4 but its ecological compensation with herbivory and predation is lower than that of 4. Our rules of dominance

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35

Table 2 Parameter values for the four different food webs Host plant

Temperature (°C)

Species

P =uD−R

R

a

Alfalfa

17

Alfalfa

21

California Bay

25

Cassava

27

Citrus

26.6

P. Aphid B. Aphid A. smithi A. er6i P. Aphid B. Aphid A. smithi A. er6i A. chilensis R. Lopanthae Mealybug A. lopezi A. di6ersicornis Ezochomus spp. Vedalia beetle C. iceryee

0.31a 0.20c 0.34d 0.28d 0.44a 0.23c 0.42d 0.34d 0.14f 0.17g 0.18h 0.17j 0.17o 0.10l 0.18m 0.14m

0.09b 0.10b 0.09b 0.09b 0.16b 0.18b 0.16b 0.16b 0.33n 0.14g 0.15i 0.15i 0.15o 0.10n 0.10n 0.12n

1b 1b 0.25b 0.25 (0.12)e 1b 1b 0.25b 0.25 (0.12)e — — 0.25i 0.65k 0.13k 0.05i — —

The data come from: a Siddiqui and Barlow (1973), b Gutierrez et al. (1990), c Summers et al. (1984), d Kambhampati and Mackauer (1989), e Bueno et al. (1993), f Pizzamiglio (1985), g Cividanes and Gutierrez (1997), h Schulthess et al. (1987), i Gutierrez et al. (1988), j Lo¨hr et al. (1989), k Gutierrez et al. (1993), l Umeh (1990). m Estimated from life history characteristics as reported in Quezada and DeBach (1973) and Boddenheimer (1951); n estimated with mass scaling law for pokiliotherms (Peters, 1983): R= e0.051T 0.055×(bodysize in mg)−1/4 mg/day, where T is temperature; o assumed to be similar to A. Iopezi.

assert: plant 1 can invade an empty resource patch excluding plant 4. The invasion of the herbivore into this patch increases plant 1’s ecological compensation point facilitating the invasion of plant 4 and resulting in the displacement of plant 1. Finally, the arrival of the predator reduces the herbivore’s growth efficiency such that its effect on either plant species is minimal. This facilitates the return of plant 1 and results in the displacement of plant 4.

regressions on developmental rates. Although the mass scaling law serves as crude estimates for respiration rates, they are reasonable when the body sizes of the compared species are significantly different (Yodzis and Innes, 1992). Since the intrinsic rates of increase, rm, equals maximal productivity, we can use the relation P= A− R= rm to estimate the value of A= uD. Table 2 summarizes the parameter values for four different food webs. The emphasis here is to determine qualitatively if the predictions of the model are in accord with the dynamics observed in the field.

4. Model predictions versus field observations

4.1. The bay tree–oleander scale system In the following four sections, we examine the population dynamics of four ecosystems using our food web model with the simplifying assumptions that the parameters are averaged over the appropriate season and developmental stages. In the absence of lab estimates, we estimate the respiration rates for a species with a mass scaling law for pokiliotherms and rescale to different temperatures using the Q10 rule (Peters, 1983) and linear

The armored scale, Aspidotus nerii (Bouche), occurs on California Bay, Umbellularia californica (Hopk. and Arn.), and it is heavily attacked by the aphelinid parasitoid, Aphytis chilensis (Howard) and the coccinellid predator, Rhysobius lophanthae (Blaisd). Pizzamiglio (1985) intensively studied these relationships in the San Francisco Bay area and showed that the coccinellid beetle

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S.J. Schreiber, A.P. Gutierrez / Ecological Modelling 106 (1998) 27–45

was by far the most important natural enemy of this scale. The thelyotokous parasitoid host feeds and is ectoparasitic on the scale and prefers to attack the larger stages. The beetle adults and larvae attack most stages but females prefer to oviposit in large pre reproductive and reproductive scales. Both natural enemies act as both predators and parasitoids and both have 2.5 generations to each generation of the scale. In general, the host finding capacity of both natural enemies is low, but the rates of consumption and fecundity of the coccinellids are high relative to the parasitoid. As R/P for R. Iopanthae and A. chilensis are :0.8 and 2.4, respectively, and both species have approximately the equivalent searching capacities (Cividanes and Gutierrez, 1997), the metabolic compensation point of the beetle is lower than the parasitoid and, hence, should displace the parasitoid in a closed system. This displacement is observed in the field (Pizzamiglio, 1985).

4.2. The citrus–cottony cushion scale system The control of the cottony cushion scale, Icerya purchasi (Maskell), is the classical example of successful biological control. As we shall see, this system has many aspects in common with the oleander system. The cottony cushion scale was accidentally introduced into California in 1868, and rapidly spread through citrus groves nearly destroying the industry. In the late 1880s, two host specific species of natural enemies were introduced from Australia; a predatory coccinellid beetle, the vedalia beetle Rodolia cardinalis (Mulsant), and a parasitic fly, Cryptochaetum iceryne (Williston). Boddenheimer (1951) and Hagen (1974) summarized the biology of the cottony cushion scale and the beetle, and Quezada and DeBach (1973) provide insights into the biology of the parasitic fly. The beetle adults and larvae attack most stages but adults prefer to oviposit in large pre-reproductive and reproductive stages, and like R. Iophanthae on oleander scale, the vedalia beetle is both a predator and a parasitoid. The beetle has four generations per generation of the scale. The fly attacks larger stages of the scale, and Quezada and DeBach suggest that it is better

at finding its host than the vedalia beetle (i.e. avedalia B afly), and it has three generations per generation of scale. Once introduced, the vedalia beetle quickly established itself and within 6 months of release rapidly reduced scale populations in most Southern California citrus groves (DeBach and Rosen, 1991), particularly in the hotter regions. The rapid control by the vedalia beetle over-shadowed the role of C. iceryae in the cooler areas of California where the parasitoid regulates scale populations to the same degree as the vedalia beetle does in hotter interior regions (Quezada and DeBach, 1973). These conclusions were confirmed by observations in the cool climate of Bermuda where the beetle failed to control the scale, whereas the fly gave excellent control (see Hagen, 1974). This suggests that climate and weather at different locations and times of the year may create conditions where one or the other natural enemy may dominate. The effect of climate on the physiology of the natural enemies is reflected in changes in the ratio of R/P for both natural enemies. For example, in the hot interior of California (see Table 2) (R/P)vedalia : 0.6B 0.9:(R/P)fly during the summer months, but this relation may be reversed during the remainder of the cooler parts of the year due to temperature effects on respiratory and developmental rates. In areas favorable for all three insect species, the parasitoid may dominate on host plants such as Cocculus laurifolius because it is not attractive to the vedalia beetle and avedalia goes to zero (Quezada and DeBach, 1973).

4.3. The African cassa6a system Using the results on persistence and estimates for the parameters reported in Table 2, we examine the cassava, Manihot esculenta (Crantz), food web in West Africa. Cassava is a primary food crop for more than 200 million people living in the African ‘cassava belt’, an area more than twice the size of the US. In 1973, the cassava mealybug, Phenacoccus manihoti (Mat.-Ferr.), was accidentally introduced causing up to 84% losses in cassava (Herren et al., 1987). A complex of native coccinellid predators of the genera Hyper-

S.J. Schreiber, A.P. Gutierrez / Ecological Modelling 106 (1998) 27–45

aspis and Exochomus attacked the mealybug but could not check pest population growth that often reached thousands per plant. In 1981, successful biological control occurred with the introduction of the exotic parasitoid Apanoggrus lopezi (De Santis). The results were spectacular, with average densities of mealybug falling to B11 per plant. In contrast, a related species Apanoggrus di6ersicornis (Howard) failed to become established despite repeated introductions (Gutierrez et al., 1993). Fig. 3a summarizes these trophic relations with C, cassava; M, mealybug; L, A. Iopezi; D, A. di6ersicornis respectively, and E representing a complex of native coccinellid predators, Ezochomus sp. and Hyperaspsis sp. The interaction between cassava, mealybug and the complex of Coccinellids (as a closed system) is highly unstable (Gutierrez et al., 1994). For the sake of discussion, we assume that the food chains, C“ M “D and C “M “L, support a globally stable equilibrium (the latter is observed in the field). To describe all possible transitions from one

Fig. 3. (a) The cassava system: cassava (C), mealybug (M), the parasitoids E. Iopezi (L) and E. di6ersicornus (D), Exochomus and Hyperaspis sp. (E). (b) The alfalfa system: alfalfa (A), pea aphid (P), blue alfalfa aphid (B), A. smithi (S), and A.er6i (E).

37

Fig. 4. The assembly diagrams for the cassava system. Abbreviations as in Fig. 3.

permanent subsystem to another, we construct assembly diagrams (Law and Morton, 1993) using the results of the analyses. These diagrams are directed graphs in which the permanent subsystems are the vertices and transitions from one permanent subsystem to another following an invasion are indicated by a directed edge. To construct the assembly diagram for the cassava system, first consider competition between the two parasitoids in the absence of the generalist. In this case, Eq. (6) implies that A. Iopezi, because of it superior searching capacity, displaces A. di6ersicornis. Hence A. Iopezi has the lower metabolic compensation point. Eq. (8) can be evaluated for the two subsystems {C, M, L, E} and {C, M, D, E}, and in both cases, the subsystems are permanent. Since predation by Ezochomus spp. is symmetric and respiration rates of both parasitoids are comparable, the ecological compensation point of A. Iopezi is lower than the ecological compensation point of A. di6ersicornis and, hence, the equilibrium attained by species {C, M, L, E} is an attractor in the C–M–D–L–E phase space and all other equilibria are not. The assembly diagram for this system is as indicated in Fig. 4. The displacement of A. di6ersicornis agrees with the field observations and is highly reasonable as A. di6ersicornis has no alternative hosts and is in temporal synchrony with A. Iopezi (i.e. the system is for all intents and purposes a closed system).

S.J. Schreiber, A.P. Gutierrez / Ecological Modelling 106 (1998) 27–45

38 Table 3 Species present

Temperature (°C)

Pea aphid’s ecp

Blue alfalfa aphid’s ecp

B, B, B, B, B, B, B, B, B, B, B, B,

17

0.31 0.51 2.71 5.04 2.46 4.40 0.38 0.52 1.36 1.82 1.27 1.70

0.52 0.55 0.52 0.55 1.81 1.89 0.77 0.80 0.77 0.80 1.73 1.79

P P, P, P, P, P, P P, P, P, P, P,

F S S, F E E, F 21 F S S, F E E, F

The ecological compensation point (ecp) of blue alfalfa aphid (B) and pea aphid (P) in the alfalfa system with different species combinations and temperatures. E, A. er6i; S, A. smithi; and F, Fungus.

4.4. The alfalfa–aphid system Alfalfa (Medicago sati6a L) is a perennial plant that is normally grown for 3 – 4 years. Approximately 1500 species occur in this system, but by and large they do not interact except via the host plant. Pea aphid, Acrthosiphon pisum (Harris), was introduced accidentally into North America in the 19th century and the blue alfalfa aphid, Acrthosiphon kondoi (Shinji), in 1974. Both species rapidly spread and became important pests of alfalfa where they may have 15 – 20 generations per year (Bueno et al., 1993). The two aphids are attacked by hymenopterous parasitoids in their native range, and hence biological control programs were begun. In 1958, Aphidius smithi (Subba Rao), a parasitoid host specific to the pea aphid, was released throughout California and led to its successful control (van den Bosch et al., 1966). In 1976, 2 years after the appearance of the blue alfalfa aphid, Aphidius er6i (Haliday), a parasitoid which attacks both pea and blue alfalfa aphid but prefers pea aphid (Bueno et al., 1993), was released in California. Upon its release, a dramatic decline in pea aphid and A. smithi populations was observed (Kambhampati and Mackauer, 1989). In addition to these species, an important component of the alfalfa crops in California is the fungal pathogen, Pandora neoaphidis R. and H., which attack pea aphid at a 10-fold

greater rate than blue alfalfa aphid (Pickering and Gutierrez, 1991). The relations between the species are summarized in Fig. 3b where A, alfalfa; P, pea aphid; B, blue alfalfa aphid, S, A. smithi and E, A. er6i. We perform the analysis with and without mortality due to the fungal pathogen. When we include fungal mortality, it is treated as a density-independent mortality factor that is activated by rain. When active, per unit biomass fungal mortality is 0.25 and 0.02 for the pea aphid and blue alfalfa aphid, respectively (Pickering and Gutierrez, 1991). Table 3 provides a summary of the ecological compensation point for the aphids when different species are active in the system. Our discussion of this system begins with an annual average temperature of 17°C. Since the pea aphid has a lower metabolic compensation point (with or without fungal mortality) than the blue alfalfa aphid, the blue alfalfa aphid can not invade in the absence of natural enemies. Introducing A. smithi or A. er6i into this subsystem raises the ecological compensation point of the pea aphid above the blue alfalfa aphid’s ecological compensation point (see Table 3). Hence, the blue alfalfa aphid can invade the system and displace the pea aphid. This concurs with the observation that the blue alfalfa aphid established itself following the introduction of A. smithi and the rapid decline of pea aphid populations prior to the

S.J. Schreiber, A.P. Gutierrez / Ecological Modelling 106 (1998) 27–45

introduction of A. er6i (Gutierrez et al., 1988). A. smithi is the dominant parasitoid species in the {A, P, S, E} system because of its slightly higher searching efficiency and P/R ratio. The fact that pea aphid fails to persist in the subsystem, {A, P, B, E}, implies that equilibrium attained by the food chain, A “B“ E, is an attractor not only in A– P – B–E phase space but also in A – P – B – E – S phase space. The resulting assembly diagram is shown in Fig. 5. Thus, our analysis argues that A. smithi is displaced not due to interspecific competition with A. er6i, rather due to a bottom-up effect (the displacement of pea aphid) that is facilitated by a top-down effect (parasitism by A. er6i and the pathogen P. neoaphidis). The dynamic trends predicted by the model agree with field observations with the following caveat. In the field, pea aphid continues to persist at extremely low numbers (Bueno et al., 1993) and may experience resurgence during years of drought. To explain this observation, we evaluate the ecological compensation point of both aphid species when the annual average temperature is 21°C. The results are summarized in Table 3. When A. er6i and the fungus are present, the difference between the ecological compensation points of the aphids is negligible and the pea aphid may or may not be able to increase. When

Fig. 5. The assembly diagram for the alfalfa system. Abbreviations as in Fig. 3.

39

we exclude the fungus, the pea aphid’s ecological compensation point is substantially lower than the blue alfalfa aphid’s ecological compensation point. Hence, the pea aphid is expected to increase in warmer weather when the fungus is not present. This prediction concurs with the fact that the pea aphid experienced a flush of growth during the last California drought (Gutierrez, 1992).

5. Discussion Although particular details of our dynamical food web study may be called into question, overall the results appeal to intuition and offer an alternative view into the mechanisms of persistence and invasion. Following the ratio-dependent trajectory through these issues helps sharpen our intuition about population dynamics by sheding light on the dynamical consequences of taking diametrically opposed views on the functional response (i.e. ratio-dependent versus purely prey dependent). By developing these and other alternative theories, we learn under what conditions the approaches agree (e.g. Eq. (6)) or not (see below). In the simplest case, a predator–prey interaction, permanence may fail due to over exploitation by the predator. This deterministic prediction has been reported before (Freedman and Mathsen, 1993) for ratio-dependent models but doesn’t occur with other functional responses (e.g. Holling types I–III, DeAngelis et al., 1975). None the less, there are many ecological systems, especially islands, that exhibit these ‘boom and bust’ phenomena (Williamson and Fitter, 1995). Examples of this include rabbits introduced to islands (Flux, 1994) and passeriforms on Hawaii (Moulton, 1993). Hence, although it has been argued that these behaviors of the model are pathological (Abrams, 1994), we agree with Akc¸akaya et al. (1995) when they state that the pathology is mathematical and not necessarily biological. When two species of consumers share a resource and a predator, we provide a simple rule that determines dominance: when predation is physiologically symmetric, the dominant consumer minimizes its ecological compensation

40

S.J. Schreiber, A.P. Gutierrez / Ecological Modelling 106 (1998) 27–45

point. These rules provide a ratio-dependent analog of the R** rule of Holt et al. (1994); if predation is completely symmetric (the predator encounters both consumers at equivalent rates and is physiologically symmetric), the dominant consumer depresses the resource to a lower equilibrium density (denoted R**) than all other consumers. Note that the R** rule reduces to the R* rule (see Section 3.2) when there is no predation. Despite this similarity, the ratio- and resource-dependent rule differ in at least two ways. First, one rule involves a consumer reducing the resource available per consumer while the other involves a consumer reducing the absolute resource density. Second, the outcome in resourcedependent rule depends on the productivity of the resource. In exploitative systems, dominance can switch from one consumer to another if resource productivity switches from low to high (Holt et al., 1994). This switch in dominance can occur because predator densities increase with resource productivity while consumer densities remain constant. For the ratio-dependent analog, this switching never occurs because resource productivity increases consumer and predator densities simultaneously. Consequently, predation costs are scale invariant. Since the systems we studied are high input agricultural systems, this potential discrepancy should not effect our results. The simple rule of dominance for ratio-dependence are characterized by four terms (maximal productivity, respiration, production efficiency and apparency) three of which provide links to the vast literature on ecological energetics (see for example, Phillipson, 1966; Petrusewicz and MacFadyen, 1970; Odum, 1971; Peters, 1983). McNeil and Lawton (1970) and Humphreys (1979) examined the relationship between annual production and respiration for terrestrial animal populations. In these studies, log – log regressions of respiration against productivity of species separated into different organismal types (e.g. noninsect invertebrate, arthropods, vertebrate ectoand endotherms) were performed. Peters (1983) also examined this relationship using mass scaling laws. All studies indicate that within an organismal type, log R =a log P +b with a close to

one (usually in a range from 0.8 to 1.1) suggesting that P/R is unaffected by changes in the absolute rate in production or respiration regardless of the variations in animal or population size (Peters, 1983). Conversely, between organismal types the differences in these ratios can be significant. For instance, Peters (1983) showed that unicells have 150-fold greater productivity to respiration ratio than homeotherms. However, since species of the same organismal classes are more likely to compete for similar resources and be subject to similar forms of predation (Cohen et al., 1993) and our results suggest that in these cases P/R plays a role in structuring communities, a clearer picture of how these relationships scale within organismal types would be useful (see McNeil and Lawton, 1970). To conclude, this simplified approach to studying food web dynamics provides a rational approach to understanding these and more complicated interactions between species. Many of the qualitative predictions of our simplified model are similar to events observed in field systems and to simulations of a detailed age structured model based on the same physiological paradigm (Gutierrez et al., 1990; Gutierrez, 1992; Gutierrez et al., 1993). Unlike these detailed simulations, many of the finer biological details are lacking and, hence, we caution against finding explanations beyond what is reasonable (see Gutierrez et al., 1994). However, there is an increasing need in ecology for short cut methods for analyzing trophodynamics and predicting the energy flow through complex communities, and we suggest that further exploration of population dynamics using physiologically based models may provide such methods. Furthermore, since changes in trophic structure are common when species are introduced accidentally or intentionally, biological control will continue to offer a fertile testing ground of these theories and point us toward species interactions that warrant further attention.

Acknowledgements We thank Don DeAngelis, Joe Eisenberg,

S.J. Schreiber, A.P. Gutierrez / Ecological Modelling 106 (1998) 27–45

Wayne Getz, Peter Lundberg and Mary Power for providing criticisms and comments that improved the quality of the manuscript, and Niels Holst for pointing out an error in the formulation of the model in an earlier incarnation of this manuscript.

mi, j =

bjiaji mj

mi,0 Di

%

bjkmk

%

mi, j,

k  Prey( j )

Since mi = mi,0 +

Appendix A

41

:

j  Prey(i )

we get that To extend the predator – prey model to n interacting species, we assume that each individual can be in any one of k + 1 states where k is the number of resources available to the individual in question. Either the individual is actively searching or has just consumed one of its k resources. We denote these various states as follows: mi,0 equals the biomass of species i that is actively searching and mi, j equals the biomass of species i that just consumed species j. Using this notation, the dynamics of these species i is given by

mi,0 = mi 1+

j  Prey(i )

%

j  Pred(i )

mi,0 mi

bjiaji mj

%

k  Pred(i )

% k  Pred( j )

:

bjkmk

bikmk

%

j  Prey(i )

:

Di mi, j −mi,0

bjiaji mj %

j  Prey( j )

bjkmk

bjiaji mj %

k  Prey( j )

Di

bijaji mj

:

%

k  Prey(i )

aki mk %

l  Pred(k)

% l  Pred(k)

bklml

;

−1

bjlml

Dlml

;

−1

aji mj %

l  Pred( j )

Dlml

Appendix B (A2) For the sake of clarity, Eq. (5) and a criteria for cones of extinction are derived for the tritrophic system given by

dmi, j =growth and respiration −Di mi, j dt + mi,0

.

When bji = Di for all species, this expression reduces to

Di 1+

;

−1

bkiaki mk

k  Prey(i )

%

(A1)

bjkmk

%

l  Pred( j )

dmi,0 = growth and respiration dt

+

%

k  Prey( j )

= 1+

·

bijaij mi

mj,0

Di

;

Thus, the functional response of species i on species j is given by

dmi = growth and respiration dt −

bjiaji mj

%

(A3)

bjkmk

where the coefficients, bjk, determine how species j is partitioned amongst the species of its consumer guild, k Pred( j ). Under the assumption that the dynamics between individual states occurs at a faster time scale than growth and respiration, Eq. (A2) and Eq. (A3) imply that individual dynamics equilibrate as follows

dm1 =u1D1m1g(a1m0/D1m1)− R1m1 − dt D2m2g(a2m1/D2m2) dm2 =u2D2m2g(a2m1/D2m2)− R2m2 − dt D3m3g(a3m2/D3m3) dm3 =u3D3m3g(a3m2/D3m3)− R3m3 dt

S.J. Schreiber, A.P. Gutierrez / Ecological Modelling 106 (1998) 27–45

42

where ai = ai − 1,i and g(x) =x/(1 + x). Suppose u1D1/(R1 + a2)B1. Then lim

m 1“0

1 dm1 u1D1g(a1R/D1m1) − R1 = mlim 1“0 m1 dt −D2g(a2m1/D2m2) ·

m2 m1

 

R3 a23m*2 = g−1 D3m*3 u3D3

] u1D1 −R1 −a2 \0. Therefore, the m1 = 0 plane is a repellor and m1 persists. On the other hand, suppose u1D1 −R1 − a2 + R2 +a3 B0. Choose K \ 0. If m2 =Km1 then

 

dm1 dm2 lim K “ ,m 1 “ 0 dt dt

−1

5K “ ,m lim

1“0

       

u1D1g

where g(x)= x/(1+ x). Thus species 4 can invade this equilibrium if and only if (m 1,m 2,m 3,m 4) “ (m *,m *,m *,0) 1 2 3

a2 −R2−a3 D2K

 

a1m0 a2 −R1−D2Kg D1m1 D2K

a2 −R2 −a3 D2K =Klim “ a2 u1D1 −R1 −D2Kg D2K − R2 −a3 = B 1. u1D1 −R1 − a2 u2D2g

(C.1)

lim

1 K

u2D2g

proof of this fact can be easily derived from stable manifold theory and the structure of these equations (see Waltman, 1991 for a discussion). To derive Eq. (6), notice the equilibrium, (m *, determined by the system in ab1 m *, 2 m *), 3 sence of species 4 satisfies

In particular, for sufficiently large K \0 and small d\ 0 the cone, C ={(m1, m2, m3): m2 5Km1 and m1 5d}, is forward invariant. Furthermore, all points in C are attracted to the origin.

\0

(C.2)

Substituting Eq. (C.1) into Eq. (C.2), this is equivalent to g



 

a24 − 1 R2 g a23 u2D2

\

R4 u4D4

Applying g − 1 to both sides, substituting g − 1(x)= x/(1−x) and rearranging terms implies species 4 invades if and only if Eq. (6) holds. The derivation of Eq. (8) starts with the obseralso savation: the equilibrium, (m *, 1 m *, 2 m *), 3 tisfies Eq. (C.1). Species 4 invades this equilibrium if and only if lim

Appendix C − Notice that

m; 4 m4

 

(m 1,m 2,m 3,m 4) “ (m *,m *,m *,0) 1 2 3

= u4D4g

The analysis necessary for Eq. (6), Eq. (8) and Eq. (9) depend on the following fact: if m; i(t) = fi (m1(t),…, mn (t)) is a population model such that each n− 1 dimensional face of the positive orthant admits a positive globally stable equilibrium (this equilibrium may not include all of the n − 1 species, but all positive initial states are attracted to it) and the boundary dynamics are acyclic, then species i persists if (fi/(mi evaluated at the unique positive equilibrium in the hyperplane: xi =0 is strictly greater than zero. On the other hand, if this derivative is less than zero, then this equilibrium is an attractor that excludes species i. The

 

m; 4 a24m*2 = u4D4g − R4 m4 D3m*3

a24m*2 − R4 D3m*3

 

a m* a43 D m*g 23 2 \ 0. a23m*2 3 3 D3m*3

(C.3)

 

a23m*2 a43 u D − R3 P = 3 3 a43 = 3 a43 D3m*g 3 D3m*3 a23m*2 u3D3 A3 Subsituting Eq. (C.1) into Eq. (C.3) and rearranging terms implies species 4 invades if and only if



 

a R3 g 24 g − 1 a23 u3D3

P3 a A3 43 u4D4

R4 + \

(C.4)

Further rearrangments of Eq. (C.4) produces Eq. (8).

43 S.J. Schreiber, A.P. Gutierrez / Ecological Modelling 106 (1998) 27–45

Finally consider the species configuration for Eq. (9). The equilibrium, (m *, satisfies 1 m *, 2 m *), 3 g

 

a23m*2 R3 = D3m*3 u3D3

and u2D2g

 

 

a12m*1 D m* a m* =R2 + 3 3 g 23 2 . D2m*2 m*2 D3m*3

m*= 1 (C.5)

Eq. (C.5) implies

 

(C.6)

Species 4 invades the food chain, 1 “2 “3, if and only if lim

= u4D4g

 

1 − 1 R2 g u2D2 a12

Hence, species 3 can invade this equilibirum if and only if u3D3g(a13m*)−R 1 3\0

a12m*1 R2 + AP33 a23 = . g D2m*2 u2D2

m; 4 m4

Equivalently, g





R a13 − 1 g (R2/u2D2) \ 3 a12 u3D3

When g(x) = x/(1+ x), we get that 3 can invade if and only if

 

(m 1,m 2,m 3,m 4) “ (m *,m *,m *,0) 1 2 3

dynamics of species 1 and all other parameters have the same interpretation as for the ratiodependent model. The equilibrium, (m *, 1 m *, 2 0), satisfies

a14m*1 P −R4 −a43 3 \0 A3 D2m*2 (C.7)

u D − R2 u3D3 − R3 a13 \ 2 2 a12 R3 R2

Subsituting Eq. (C.6) into Eq. (C.7), we get





a14 − 1 R2 + AP33 a23 g g a12 u2D2



R + P3 a \ 4 A 3 43. u4D4

Rearranging terms results in Eq. (9). The derivation of Eq. (11) is similar to Eq. (9) except the effect production efficiency of the herbivore (species 2) is modified by the costs of predation at equilibirum.

Appendix D A standard model of exploitative competition is given by a12m1 dm1 =f(m1) m1 −D2m2 dt 1 + a12m1 −D3m3

a13m1 1+ a13m1

dm2 a12m1 =u2D2m2 −R2m2 dt 1+ a12m1 dm3 a13m1 =u3D3m3 −R3m3 dt 1+ a13m1 and f(m1) represent the growth and respiration

References Abrams, P.A., 1994. The fallacies of ratio-dependent predation. Ecology 75, 1842 – 1850. Akc¸akaya, H.R., Arditi, R., Ginzburg, L.R., 1995. Ratio-dependent predation: an abstraction that works. Ecology 76, 995 – 1004. Arditi, R., Ginzburg, L.R., 1989. Coupling in predator-prey dynamics: ratio-dependence. J. Theor. Biol. 139, 311 – 326. Arditi, R., Michalski, J., 1995. Nonlinear food web models and their responses to increased basal productivity. In: Polis, G.A., Winemiller, K.O. (Eds), Food Webs: Integration of Patterns and Dynamics. Chapman and Hall, London. Begon, M., Harper, J.L., Townsend, C.R., 1990. Ecology: Individuals, Populations and Communities. Blackwell, Boston. Berryman, A.A., 1992. The origins and evolution of predator – prey theory. Ecology 73, 1530 – 1537. Berryman, A.A., Michalski, J., Gutierrez, A.P., Arditi, R., 1995. Logistic theory of food web dynamics. Ecology 76, 336 – 343. Boddenheimer, F.S., 1951. Citrus entomology in the Middle East with special reference to Egypt, Iran, Irak, Palestine, Syria and Turkey. Vitgeverij. Dr. Junk. Gravenhage pp. 663. Bueno, V.H.P., Gutierrez, A.P., Ruggle, P., 1993. Parasitism by Aphidius er6i (Hym: Aphidiidae): preference for pea aphid and blue alfalfa aphid (Hoary: Aphididae) and competition with A. smithi ). Entomophaga 38, 273 – 284.

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